P versus NP Math 40210, Spring 2012 September 16, 2012 Math 40210 (Spring 2012) P versus NP September 16, 2012 1 / 9
Properties of graphs A property of a graph is anything that can be described without referring to specific vertices Examples of properties: being bipartite having an Euler circuit having a Hamilton cycle Examples of non-properties: vertices 1, 2 and 3 form a triangle vertices u and v are in different components Math 40210 (Spring 2012) P versus NP September 16, 2012 2 / 9
The decision problem for a property The decision problem for a particular property is to determine, for any possible input graph G, whether or not G has the property Examples of decision problems: Is G bipartite? Does G have an Euler circuit? Is G Hamiltonian? Math 40210 (Spring 2012) P versus NP September 16, 2012 3 / 9
The class P A property is in class P if there is a quick procedure for answering the decision problem, that works for every possible input graph (here quick formally means that the number of steps that the procedure takes is at most a polynomial in the number of vertices of the graph) Examples of P properties: Being bipartite: pick a vertex to put in X; put its neighbors in Y, put the neighbors of these new vertices in X, etc.; wait until either the process finishes successfully (in which case G is bipartite), or it breaks down (in which case G has an odd cycle and is not bipartite) Having an Euler circuit: look at the vertex degrees; if they are all even, then G has an Euler circuit; if one or more is odd, then it does not P stands for polynomial time Math 40210 (Spring 2012) P versus NP September 16, 2012 4 / 9
The class NP A property is in class NP if, for every graph G for which the answer to the decision problem is YES, it is possible to present a quick proof of this fact (here again, quick formally means that the number of steps that need to be taken to verify that the proof is correct is at most a polynomial in the number of vertices of the graph) Important note: the proof that G has the property has to be short, but there s no limit to the amount of time needed to come up with the proof Another way to say this: a property is in class NP if whenever a graph G has the property, I can quickly convince you that it has the property, as long as I am given as much time as I need to prepare before beginning to convince you NP stands for nondeterministic polynomial time Math 40210 (Spring 2012) P versus NP September 16, 2012 5 / 9
Examples of NP properties Being bipartite: Before talking to you I find a valid bipartition X Y, and I show it to you. Or: while you watch, I run the algorithm that answers the decision problem Having an Euler circuit: Before talking to you I find an Euler circuit, and I show it to you. Or: while you watch, I run the algorithm that answers the decision problem Any P property is also NP: to convince you that G has the property, I run the (quick) algorithm that answers the decision problem Having a Hamilton cycle: Before talking to you, I find a Hamilton cycle in the graph (this may take a very long time); once I have found it, I show it to you, and clearly you can quickly verify that it is indeed a Hamilton cycle Math 40210 (Spring 2012) P versus NP September 16, 2012 6 / 9
The $1,000,000 question P: properties for which the decision problem can be quickly solved NP: properties for which a YES answer to the decision problem can be quickly verified, given enough preparation time We ve seen that P NP. The $1,000,000 question is this: Is P = NP? In other words, is it true that every decision problem for which a YES answer can be quickly verified, can also be quickly solved? A specific example: is there a quick way of deciding whether a given graph has a Hamilton cycle? Math 40210 (Spring 2012) P versus NP September 16, 2012 7 / 9
The class NP-complete A property is in the class NP-complete if a procedure for solving the decision problem for that property can be converted into a procedure for solving the decision problem for any other NP property, without any significant slow down NP-complete properties are in a sense the hardest properties: if you solve the decision problem for any one of them, you ve solved the decision problem for all other NP properties Having a Hamilton cycle is known to be an NP-complete property. So: if you find a quick way to answer the question does G have a Hamilton cycle, you ve shown P = NP if you prove that no such quick way exists, you ve shown P NP Computers and Intractability: A Guide to the Theory of NP-Completeness by Garey and Johnson (1979, W. H. Freeman publisher) 350 pages of NP-complete problems Math 40210 (Spring 2012) P versus NP September 16, 2012 8 / 9
The Millennium Prize Problems In 2000, the Clay Mathematics Institute identified seven important open problems in mathematics, and offered a prize of $ 1,000,000 for a solution to each one; see http://www.claymath.org/millennium/. One of these seven is the P versus NP problem The other six are problems of the form prove X, with the $1,000,000 only being offered for a proof of X, not a counterexample. For P versus NP, the full prize is guaranteed, whichever way the problem is resolved Math 40210 (Spring 2012) P versus NP September 16, 2012 9 / 9